This invention deals with a particular kind of heterogeneous system, which can be described as a porous body consisting of a continuous solid matrix with embedded pores that can be filled with either gas or liquid. According to S. Lowell et al, the spatial distribution between the solid matrix and pores can be characterized in terms of a porosity and pore size. According to J. Frenkel and M. A. Biot, the mechanical properties of such systems with respect to any applied oscillating stress depend primarily on the visco-elastic properties of the matrix. Lyklema notes that when such porous bodies are saturated with liquid, additional properties are then related to any surface charge on these pores, which in turn is commonly characterized by a zeta potential. Although methods exist for characterizing these mechanical and electrical properties, they all have limitations and call out for improvement.
For example, S. Lowell et al describe in details several methods for characterizing porosity and pore size. According to IUPAC pores are classified into three classes: micropores, pore size <2 nm; mesopores, pore size is between 2 and 50 nm; macropores, pore size >50 nm. Gas adsorption techniques are applied for micro- and mesopores analysis. Mercury porosimetry has been the standard technique for macropore analysis. Environmental concerns justify a search for alternative methods that might eliminate, or at least minimize, the use of this dangerous material.
Lyklema and a IUPAC Report describe a method for measuring the zeta potential of porous bodies using the method of streaming current/potential. However, this method is not applicable for characterizing ζ-potential of small pores due to the low hydrodynamic permeability of these pores, and furthermore it is not suitable for simultaneously determining porosity and pore size.
There have been several attempts at developing ultrasound methods for characterizing porous bodies. A sound wave undergoes change as it propagates though a saturated porous body and, in the process, generates a host of secondary effects that can then be used for characterizing the properties of these bodies. To date, most attempts are associated with the measurement of sound speed and attenuation, the two main characteristics of ultrasound waves propagating through a visco-elastic media. These two parameters are easily measurable and, in principle, can serve as a source of information for calculating porosity and pore size. U.S. Pat. No. 6,684,701 issued Feb. 3, 2004, to Dubois et al describes a method for extracting porosity by comparing the measured attenuation spectra with that of predetermined standards. U.S. Pat. No. 6,745,628, issued Jun. 8, 2004, to Wunderer claims to measure porosity based on transmission measurements of ultrasonic waves in air, which might be possible only for very large pores comparable to the sound wavelength, which for proposed low frequency is perhaps several millimeters. Yet another U.S. Pat. No. 7,353,709, issued Apr. 8, 2008, to Kruger et al., suggests some improvements in this method, but still relies on comparison with attenuation standards to extract the porosity information from the raw data. There are also several patents describing the use of ultrasound for characterizing the porous structure of bone. One example is U.S. Pat. No. 6,899,680, issued May 31, 2005, to Hoff et al. for estimating the shear wave velocity, but not attenuation. There are also two patents that utilize differences in sound speed between different propagation modes. The first is U.S. Pat. No. 5,804,727, issued Sep. 8, 1998, to Lu et al., that simply states that a person skilled in the art would recognize that velocities of different modes could be used for determining the physical properties of materials. The second, U.S. Pat. No. 6,959,602, issued Nov. 1, 2005, to Peterson et al., suggests that, based on a prediction by Biot, one might use the velocity of fast compression waves for calculating porosity and slow compression waves for detecting body defects.
However, analysis of the Biot theory raises many concerns about the efficacy of using ultrasound attenuation and sound speed for characterizing porous bodies. M. A. Biot in 1956, crediting the earlier work by J. Frenkel, developed a well-known general theory of sound propagation through wet porous bodies by including the following set of eleven physical properties to describe the solid matrix and liquid:
1. density of sediment grains
2. bulk modulus of grains
3. density of pore fluid
4. bulk modulus of pore fluid
5. viscosity of pore fluid
6. porosity
7. pore size parameter
8. dynamic permeability
9. structure factor
10. complex shear modulus of frame
11. complex bulk modulus of frame
Ogushwitz recognized that the last four of these properties present a big problem in applying Biot's theory and proposed several empirical and semi-empirical methods for estimating their value, but none of his suggestions are sufficiently general, and in some cases simply amount to a substitution of one property with another unknown constant. Barret-Gultepe et al also discuss this problem in their study of the compressibility of colloids, in which they speak of the importance of a “skeleton effect” and the difficulty of measuring the required input parameters independently.
This problem of unknown input parameters makes us skeptical of determining porosity and pore sizes from attenuation and sound speed. Furthermore, relying on attenuation and sound speed alone does not yield any information on the electric properties of porous bodies. However, the electric properties can be determined using ultrasound since sound generates electric signals as the sound wave propagates through the porous body by disturbing the electric double layer surrounding the pore surfaces. This effect is usually called the “seismo-electric phenomena” because it was first employed in the field of geological exploration. Ivanov first discovered the effect in 1940 and J. Frenkel developed the first relevant theory in 1944. Independently, M. Williams in 1948 discovered the same effect and later in 2007, A. Dukhin described this effect in a dispersion of structured carbon nano-tubes. Each used different names for essentially the same effect. Apparently, E. Muller et al. also observed this effect in chromatographic resins, but mistakenly interpreted it as a Colloid Vibration Current, as described in detail by A. Dukhin and P. Goetz. There is a host of papers on the observation of this seismo-electric effect in geology including M. G. Markov in 2004, Z. Zhu et al. in 2007 and 1999, S. R. Pride et al. in 1994 and 1996, M. W. Haarsten et al. in 1997, Mikhailov et al. in 2000, and A. Thomson et al. in 1993.
This high frequency effect is very close in nature to streaming current/potential, which is also the result of coupling between mechanical and electric fields. However, there is one large difference between this seismo-electric current and a streaming current. Streaming current is typically measured under steady state or very low frequency excitation. Consequently, it is an isochoric effect, which means that the liquid and the solid matrix are considered incompressible. In contrast, the seismo-electric effect is non-isochoric by its very nature. A relative motion of phases in pores occurs due to a difference in compressibility between the liquid and solid. This difference justifies using a new independent name for this effect. Seismo-electric and reverse electro-seismic effects belong to a family of electro-acoustic phenomena, which was described in some detail by A. S. Dukhin and P. J. Goetz in 2002. These phenomena are each associated with coupling between electric and ultrasound fields in heterogeneous systems. Nevertheless, there are many similarities between these effects, which is why an overview of streaming current/potential features is important part of this prior art discussion.
It is well known that a flow of liquid through a porous body generates an electric current or potential, depending on the method of measurement. This electric response occurs due to the motion of the ionic diffuse layers that screen the electric surface charge covering the pores. This motion would appear with an applied mechanical driving force at any frequency. However, the mechanism is quite different for a constant or low frequency driving force, as compared with high frequency excitation at several MHz or higher. This difference in applied frequency provides some justification for using two different terms for essentially the same electro-mechanical effect.
Historically, the first experimental observation of this coupling in porous bodies was made for a constant applied driving force. In the field of Colloid and Interface Science, this effect is known as streaming current or streaming potential. It is usually assumed that the gradient of the applied pressure is constant, time independent, and that the liquid is incompressible, which makes this an isochoric phenomenon. M. W. Kruyt in 1952, S. Dukhin et al. in 1974 and J. Lyklema in 2000 discuss many experimental and theoretical studies of this effect.
Streaming current/potential depends strongly on the distribution of the electric potential inside the pores Φ. FIG. 1 illustrates possible space distributions of this potential including two extreme cases: (1) isolated thin double layer (DL) and (2) homogeneous completely overlapped double layers. There are analytical theories that describe the main features of this electro-kinetic effect for these two extreme cases.
M. Smoluchowski in 1903 was the first to develop the well-known theory for streaming current/potential for the case of isolated thin double layers. This theory yields the following expression for the electric potential difference ΔV generated by pressure difference ΔP:
                              Δ          ⁢                                          ⁢          V                =                                            ɛ              ⁢                                                          ⁢                              ɛ                0                            ⁢              ζ                                      η              ⁢                                                          ⁢                              K                m                                              ⁢          Δ          ⁢                                          ⁢          P                                    (        1        )            
where ∈ and ∈0 are the dielectric permittivity of the liquid and vacuum, ζ is the electro-kinetic potential of the pore surface, η is the dynamic viscosity of the liquid, and Km is the conductivity of the liquid.
This theory is valid when the capillary radius R is much larger then the DL thickness, κ−1, i.e.:κR>>1  (2)
as illustrated by curve 1 on FIG. 1.
It is also assumed that surface conductivity associated with excess ions in the DL is negligible.
According to Smoluchowski theory, this effect offers little hope for studying porosity and pore size, since these parameters are simply absent in Eq. 1. This is an unfortunate result of the geometric similarity between the hydrodynamic and electric fields in the pores under conditions where Smoluchowski theory is valid. The introduction of surface conductivity would introduce some dependence on pore size, but even this is uncertain because of the unknown values for the surface conductivity.
Decreasing the pore size leads eventually to an overlap of the double layers, which is reflected as a transition from distribution 1 to distributions 3 on FIG. 1. Further decreasing of the pore size would cause complete overlap and the electric potential in the pore becomes constant. This is the case of the “thick” and “homogeneous” double layer, when:κR<<1  (3)
It is curious that the theory corresponding to this case of complete overlap was developed not in field of general Colloid Science but by scientists dealing with membrane phenomena, such as reverse osmosis and hyperfiltration. The reason for this is that condition (3) is valid in water only for very small pore size, which of course is the case for reverse osmosis membranes. This theory is made quite complex due to “concentration polarization”, a phenomenon that occurs in sufficiently charged membranes when the number of counter-ions substantially exceeds the number of co-ions in the pores. This concentration polarization leads to a separation of ions at the pore entrance, which in turn generates a concentration gradient in front of pore. G. B. Tanny and E. Hoffer in 1973 developed a theory of streaming current/potential for the “homogeneous” case that takes into account this concentration polarization. We present here just the final expression to illustrate the complexity of this theory:
                              FE          RT                =                                                                              J                  v                                ⁢                                  f                                      1                    ⁢                    w                                    0                                                            θ                ⁢                                                                  ⁢                                  φ                  w                                                      ⁢            Δ            ⁢                                                  ⁢            x                    +                      ln            ⁢                                                  ⁢                                                                                c                    ~                                    s                  ″                                +                X                                                                                  c                    ~                                    s                  ′                                +                X                                              -                                    1              2                        ⁢            ln            ⁢                                                  ⁢                                                                                                      c                      ~                                        s                    ″2                                    +                                                                                    c                        ~                                            s                      ″                                        ⁡                                          (                                              X                        -                                                                                                            c                              ~                                                        s                            ″                                                    ⁢                                                      φ                            w                                                                                              )                                                        -                                                                                    c                        ~                                            s                      ″                                        ⁢                                          t                      1                                        ⁢                    X                    ⁢                                                                                  ⁢                                          φ                      w                                                                                                                                  c                      ~                                                              s                      ⁢                                                                                                                                  ′                      ⁢                                                                                          ⁢                      2                                                        +                                                                                    c                        ~                                            s                      ′                                        ⁡                                          (                                              X                        -                                                                              c                            s                            ″                                                    ⁢                                                      φ                            w                                                                                              )                                                        -                                                                                    c                        ~                                            s                      ″                                        ⁢                                          t                      1                                        ⁢                    X                    ⁢                                                                                  ⁢                                          φ                      w                                                                                  ++                        ⁢                                          X                +                                                                            c                      ~                                        s                    ″                                    ⁢                                                            φ                      w                                        ⁡                                          (                                              1                        -                                                  4                          ⁢                                                      t                            2                                                                                              )                                                                                                  2                ⁢                C                                      ⁢            ln            ⁢                                                            {                                                            2                      ⁢                                                                        c                          ~                                                s                        ″                                                              +                    X                    -                                                                                            c                          ~                                                s                        ″                                            ⁢                                              φ                        w                                                              -                    C                                    }                                ⁢                                  {                                                            2                      ⁢                                                                        c                          ~                                                s                        ′                                                              +                    X                    -                                                                                            c                          ~                                                s                        ″                                            ⁢                                              φ                        w                                                              +                    C                                    }                                                                              {                                                            2                      ⁢                                                                        c                          ~                                                s                        ″                                                              +                    X                    -                                                                                            c                          ~                                                s                        ″                                            ⁢                                              φ                        w                                                              +                    C                                    }                                ⁢                                  {                                                            2                      ⁢                                                                        c                          ~                                                s                        ′                                                              +                    X                    -                                                                                            c                          ~                                                s                        ″                                            ⁢                                              φ                        w                                                              -                    C                                    }                                                                                        (        4        )            
where E is electromotive force of streaming effect, R is gas constant, T is absolute temperature, F is Faraday constant, X is effective charge density, Jv is volume flow, fij is friction factor between species i and j, θ is tortuosity coefficient, φw is water fraction of water in membrane, Δx is thickness of the membrane, {tilde over (c)}s is local salt concentration.
This Tanny-Hoffer theory clearly indicates the complexity of the concentration polarization phenomena. In order to use this theory for characterizing pores, it would be necessary to find experimental conditions where this concentration polarization would not develop. Practically speaking, the only available solution is to use an alternating driving force instead of a constant excitation, as is typical in traditional streaming current/potential measurements. To this end, there are several theoretical and experimental studies of the streaming current under oscillating pressure conditions. R. G. Packard in 1953, C. E. Cooke in 1955, J. N. Groves et al. in 1975, S. S. Dukhin et al. in 1983 and 1984, and L. Renaud et al. in 2004 and U.S. Pat. No. 3,917,451, issued in 1975, to J. N. Groves and J. H. Kaplan all speak to measurements under oscillatory conditions. Their theoretical analysis indicates the appearance of the additional multiplier, which depends on Bessel functions of kR, but no additional dependence on porosity.
Essentially all these studies assume isochoric conditions and consequently are not directly applicable for describing the electrokinetic effect when isochoric conditions are not valid. This assumption was justified in the above documents because rather low frequencies excitation was applied. However, the use of low frequency excitation in the KHz range does not prevent formation of concentration polarization. Instead, the frequency must be in MHz range for complete elimination of this complex effect. However, isochoric condition does not hold at such high frequency and instead of streaming current, we should deal with seismo-electric phenomena.
J. Frenkel developed the first theory of the seismo-electric phenomena in 1944. He used Smoluchowski theory of the streaming current as a starting point, which is why his theory is valid only for the case of isolated thin DL. He derived the following expression for the electric field strength E induced by ultrasound in a porous body saturated with water:
                    E        =                                            8              ⁢              ɛ              ⁢                                                          ⁢                              ɛ                0                            ⁢              ζ              ⁢                                                          ⁢              k              ⁢                                                          ⁢                              ω                2                            ⁢              f              ⁢                                                          ⁢                              ρ                m                                                    η              ⁢                                                          ⁢                              K                m                            ⁢                              r                2                                              ⁢                      (                                                                                K                    2                                                        ρ                    m                                                  ⁢                                  β                                                            β                      ′                                        ⁢                                          c                      s                      2                                                                                  -              1                        )                    ⁢          u                                    (        5        )            
where u is amplitude of displacement, cs is sound speed, K2 is compressibility modulus of the liquid, ρn, is density of liquid, f is porosity, k is Darcy constant that is proportional to the square of the pores radius r,
      β    =          1              f        ⁡                  (                      1            +            α                    )                      ;            β      ′        =          1      +                        (                      β            -            1                    )                ⁢                              K            2                                K            0                              
where K0 is compressibility modulus of the solid phase, α is coefficient proportionality between variations of volume of solid and liquid phases.
Frenkel predicted that the electro-seismic electric field strength is proportional to porosity and independent of pore size, because of the Darcy constant dependence on the square of the pore size that cancels out size dependence in Eq. 5. There is no conclusive experimental confirmation of this prediction so far. This conclusion is valid only for isolated and thin double layers, which corresponds to the electric potential distribution in pores as illustrated by curve 1 on FIG. 1.
It is quite possible that the electro-seismic effect for the “homogeneous” completely overlapped double layer, case 2 on FIG. 1, would become dependent on pore size. This would open the possibility to characterize both pores size and porosity using two different liquids for saturating the porous body. Unfortunately, there is no theory valid for the “homogeneous” case yet.
Frenkel's equation does not present E as a function of the pressure gradient, in contrast with modern electro-acoustic theories. It also ignores phase shift in the electro-seismic signal that occur at ultrasonic frequencies. Existing theoretical developments in this field, instead of following the lines drawn by Frenkel 60 years ago, have shifted emphasis to the structure of the acoustic field in the soil. However, they ignore a basic principle underlying all electrokinetic theories at Smoluchowski limit regarding the similarity of the space distribution of hydrodynamic and electrodynamic fields in electrokinetics.
Despite the lack of adequate theories, the electro-seismic effect is very promising for characterizing porous bodies since it offers four important advantages:                1. the ability to characterize very narrow pores in bodies with very low permeability, which complicates or even prevents pumping liquid in continuous flow mode;        2. the elimination of concentration polarization;        3. the possibility to characterize porosity and pore size using two different wetting liquids;        4. the simultaneous characterization of electric properties of the pores surfaces.        
U.S. Pat. No. 7,340,348, issued in March 2008, to Strack et al. discusses the acquisition and interpretation of seismo-electric and electro-seismic data, but is dedicated to empirical characterization of geological structures and makes no claims about the properties of pores.
Rather than use the seismo-electric effect on a geologic scale, we propose using it on a smaller laboratory scale, which we can achieve using a commercially available electro-acoustic device described in U.S. Pat. No. 6,449,563, issued September 2002 to A. S. Dukhin and P. J. Goetz. This instrument launches ultrasound pulses into a heterogeneous system and measures the electric response. According to said patent, the claimed use of the device was for characterizing particle size and zeta potential of dispersions and emulsions. In contrast, the current invention proposes using essentially the same device for characterizing the porosity, pore size and zeta potential of porous bodies, employing some modifications in sample handling and calibration. This porosity and pore size characterization is possible because the measured electro-acoustic signal generated by ultrasound in porous bodies is essentially seismo-electric current generated on scale of laboratory device.